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RiemannSieltjes vs. Lebesgue 
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#1
Dec412, 08:56 PM

P: 14

Hey guys,
I'm doing a paper on the Radon transform and several sources I've come across cite the Lebesgue integral as a necessary tool to handle measures in higher order transforms. But, Radon's original paper employs the RiemannStieltjes integral in its place. I read that Lebesgue is more general and so Radon could have used it in place of RSI. Is this the case? Thanks, Jeff 


#2
Dec512, 08:07 AM

Mentor
P: 18,036

The Lebesgue integral is indeed more general than the Riemann integral.
Using measure theory, we can also develop the LebesgueStieltjes integral, and this is a generalization of the RiemannStieltjes integral. So yes, the paper could probably be written with Lebesgue instead of Riemann. But there may be technical differences between the two. 


#3
Dec512, 10:15 AM

Sci Advisor
P: 3,252

Is there a distinction between "Lebesgue Integration" and "integration with respect to Lebesgue measure"? My impressions is that "Lebesgue measure" on the real number line is a particular measure that implements the usual notion of length, so the measure of a single point would be zero. On the other hand, is "Lebesgue integration" defined with respect to an arbitrary measure?
For Lebesgue Integration to include RiemannStieljes integration as a special case, is it necessary to use measures other than Legesgue measure? (I'm thinking of the specific example of defining an integration that can integrate a discrete probability density function by the method of assigning nonzero measure to certain isolated points and turning "integration" into summation.) 


#4
Dec512, 03:10 PM

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P: 6,038

RiemannSieltjes vs. Lebesgue
LebesgueStieljes integral is best described as Lebesgue integration with respect to a given measure.



#5
Dec512, 06:14 PM

Sci Advisor
P: 820

The Stieljes measure is derived from the set function ##m(a,b]) = g(b)g(a)## for some monotonically increasing function g. 


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